Quantum Channel Capacity

The quantum capacity Q of a channel N is the maximum rate, in qubits per channel use, at which quantum information can be reliably transmitted using that channel asymptotically many times with vanishing error. It is characterized by the coherent information I_c(rho, N) = S(N(rho)) - S((id otimes N)(|psi><psi|)), where |psi> is a purification of rho and the second term is the entropy of the channel’s output jointly with the reference system. The quantum capacity is Q = lim_{n->inf} (1/n) max_rho I_c(rho^{otimes n}, N^{otimes n}), a regularized formula that in general cannot be simplified to a single-letter expression.

Classical capacity C of a quantum channel describes the maximum rate of reliable classical bit transmission and is governed by the Holevo chi quantity chi(N, {p_i, rho_i}) = S(N(sum p_i rho_i)) - sum p_i S(N(rho_i)). The Holevo-Schumacher-Westmoreland theorem establishes that C = lim_{n->inf} (1/n) chi^(N^{otimes n}), where chi^ maximizes over input ensembles. A third capacity, the entanglement-assisted classical capacity C_E = max_rho I(rho, N) where I is the mutual information, gives the classical bit rate when unlimited shared entanglement is available. Remarkably, C_E admits a single-letter formula and equals the quantum mutual information, making it the most tractable of the three.

Superadditivity is a defining and practically inconvenient feature of quantum capacities: the capacity of n copies of a channel can strictly exceed n times the single-copy capacity. This means Q(N^{otimes n}) > n Q(N) for some channels, so the single-copy coherent information underestimates the true capacity. The most striking illustration is the quantum erasure channel at the threshold erasure probability and entanglement-breaking channels, but superadditivity has been proven for depolarizing channels as well. As a consequence, computing the exact quantum capacity of common noise models such as the depolarizing channel remains an open problem, with only upper and lower bounds available.

Quantum channel capacities directly inform the design of quantum repeater networks. A fiber segment with transmittance eta has a secret key capacity bounded by -log(1 - eta) bits per mode, limiting how far a single optical fiber can reliably carry quantum information without repeaters. Quantum repeaters subdivide long links into shorter segments with higher transmittance, storing and entangling quantum states at intermediate nodes to effectively boost the end-to-end capacity. The entanglement-assisted capacity C_E is relevant when pre-distributed Bell pairs are available across links, as in satellite-based quantum networks. Understanding these capacity limits guides decisions about repeater spacing, quantum memory requirements, and error correction overhead in next-generation quantum internet infrastructure.